Once the inverse rule is constructed, it is easy to define inverse function. However, we should be careful in one important aspect. An inverse function, “
f
-
1
f
-
1
” is a function ultimately. This puts the requirement that every element of the domain of the new function “
f
-
1
f
-
1
” should be related to exactly one element to its co-domain set.
We must also understand that this new function, “
f
-
1
f
-
1
”, gives the perspective of relation from co-domain to domain of the given function “f”. However, new function “
f
-
1
f
-
1
” is read from its new domain to its new co-domain. After all this is how a function is read. This simply means that domain and co-domain of the function “f” is exchanged for “
f
-
1
f
-
1
”.
Further, inverse function is inverse of a given function. Again by definition, every element of domain set of the given function “f” is also related to exactly one element of in its co-domain. Thus, there is bidirectional requirement that elements of one set are related to exactly one element of other set. Clearly, this requirement needs to be fulfilled, before we can define inverse function.
In other words, we can define inverse function, "
f
-
1
f
-
1
", only if the given function is an injection and surjection function (map or relation) at the same time. Hence, iff function, “f” is a bijection, then inverse function is defined as :
f
-
1
:
A
→
B
by
f
-
1
x
for all
x
∈
A
f
-
1
:
A
→
B
by
f
-
1
x
for all
x
∈
A
We should again emphasize here that sets “A” and “B” are the domain and co-domain respectively of the inverse function. These sets have exchanged their place with respect to function “f”. This aspect can be easily understood with an illustration. Let a function “f” , which is a bijection, be defined as :
Let A = {1,2,3,4} and B={3,6,9,12}
f
:
A
→
B
by
f
x
=
3
x
for all
x
∈
A
f
:
A
→
B
by
f
x
=
3
x
for all
x
∈
A
Then, the function set in the roaster form is :
⇒
f
=
{
1,3
,
2,6
,
3,9
,
4,12
}
⇒
f
=
{
1,3
,
2,6
,
3,9
,
4,12
}
This function is clearly a bijection as only distinct elements of two sets are paired. Its domain and co-domains are :
⇒
Domain of “f”
=
{
1,2,3,4
}
⇒
Domain of “f”
=
{
1,2,3,4
}
⇒
Co-domain of “f”
=
{
3,6,9,12
}
⇒
Co-domain of “f”
=
{
3,6,9,12
}
Now, the inverse function is given by :
f
-
1
:
A
→
B
b
y
f
x
=
x
3
for all
x
∈
A
f
-
1
:
A
→
B
b
y
f
x
=
x
3
for all
x
∈
A
where
A
=
{
3,6,9,12
}
where
A
=
{
3,6,9,12
}
In the roaster form, the inverse function is :
⇒
f
-
1
=
{
3,1
,
6,2
,
9,3
,
12,4
}
⇒
f
-
1
=
{
3,1
,
6,2
,
9,3
,
12,4
}
Note that we can find inverse relation by merely exchanging positions of elements in the ordered pairs. The domain and co-domain of new function “
f
-
1
f
-
1
” are :
⇒
Domain of
f
-
1
=
{
3,6,9,12
}
⇒
Domain of
f
-
1
=
{
3,6,9,12
}
⇒
Co−domain of
f
-
1
=
{
1,2,3,4
}
⇒
Co−domain of
f
-
1
=
{
1,2,3,4
}
Thus, we see that the domain of inverse function “
f
-
1
f
-
1
” is co-domain of the function “f” and co-domain of inverse function “
f
-
1
f
-
1
“ is domain of the function “f”.
